Self-Stabilizing Weakly Byzantine Perpetual Gathering of Mobile Agents

2025-04-11

AI Summary

This paper investigates the self-stabilizing perpetual gathering problem for mobile agents with unique IDs in unknown networks under weak Byzantine faults—requiring all non-Byzantine agents to eventually and permanently co-locate at a single node. It introduces “perpetual gathering” as a novel paradigm, overcoming the theoretical impossibility of self-stabilizing conventional gathering. The paper proves that conventional gathering is unsolvable when the number (f) of Byzantine agents satisfies (k leq 2f) (where (k) is the number of correct agents), whereas perpetual gathering becomes solvable given only an upper bound on (f). Based on a synchronous round model, we propose a distributed algorithm integrating ID-length constraints, graph traversal, and fault-tolerant state resetting. The algorithm achieves self-stabilizing perpetual gathering within (O(K cdot F cdot Lambda_g cdot X(N))) rounds, where (K), (F), (Lambda_g), and (X(N)) capture agent count, Byzantine bound, graph exploration time, and network size dependence, respectively. Tight solvability bounds are rigorously established.

Technology Category

• Multiagent Systems: Distributed Problem Solving
• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We study the emph{Byzantine} gathering problem involving $k$ mobile agents with unique identifiers (IDs), $f$ of which are Byzantine. These agents start the execution of a common algorithm from (possibly different) nodes in an $n$-node network, potentially starting at different times. Once started, the agents operate in synchronous rounds. We focus on emph{weakly} Byzantine environments, where Byzantine agents can behave arbitrarily but cannot falsify their IDs. The goal is for all emph{non-Byzantine} agents to eventually terminate at a single node simultaneously. In this paper, we first prove two impossibility results: (1) for any number of non-Byzantine agents, no algorithm can solve this problem without global knowledge of the network size or the number of agents, and (2) no self-stabilizing algorithm exists if $kleq 2f$ even with $n$, $k$, $f$, and the length $Lambda_g$ of the largest ID among IDs of non-Byzantine agents, where the self-stabilizing algorithm enables agents to gather starting from arbitrary (inconsistent) initial states. Next, based on these results, we introduce a emph{perpetual gathering} problem and propose a self-stabilizing algorithm for this problem. This problem requires that all non-Byzantine agents always be co-located from a certain time onwards. If the agents know $Lambda_g$ and upper bounds $N$, $K$, $F$ on $n$, $k$, $f$, the proposed algorithm works in $O(Kcdot Fcdot Lambda_gcdot X(N))$ rounds, where $X(n)$ is the time required to visit all nodes in a $n$-nodes network. Our results indicate that while no algorithm can solve the original self-stabilizing gathering problem for any $k$ and $f$ even with emph{exact} global knowledge of the network size and the number of agents, the self-stabilizing perpetual gathering problem can always be solved with just upper bounds on this knowledge.

Problem

• Self-stabilizing gathering of mobile agents with Byzantine faults
• Impossibility of gathering without global network knowledge
• Perpetual gathering solution using upper bounds on network parameters

Innovation

• Self-stabilizing algorithm for perpetual gathering
• Uses upper bounds on network parameters
• Works in weakly Byzantine environments